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Theorem nobndlem2 31556
 Description: Lemma for nobndup 31563 and nobnddown 31564. Show a particular abstraction is an ordinal. (Contributed by Scott Fenton, 17-Aug-2011.)
Hypotheses
Ref Expression
nobndlem2.1 𝑋 ∈ {1𝑜, 2𝑜}
nobndlem2.2 𝐶 = {𝑎 ∈ On ∣ ∀𝑛𝐹𝑏𝑎 (𝑛𝑏) ≠ 𝑋}
Assertion
Ref Expression
nobndlem2 ((𝐹 No 𝐹𝐴) → 𝐶 ∈ On)
Distinct variable groups:   𝐹,𝑎,𝑏,𝑛   𝑋,𝑎,𝑏
Allowed substitution hints:   𝐴(𝑛,𝑎,𝑏)   𝐶(𝑛,𝑎,𝑏)   𝑋(𝑛)

Proof of Theorem nobndlem2
StepHypRef Expression
1 nobndlem2.2 . 2 𝐶 = {𝑎 ∈ On ∣ ∀𝑛𝐹𝑏𝑎 (𝑛𝑏) ≠ 𝑋}
2 nobndlem1 31555 . . . 4 (𝐹𝐴 → suc ( bday 𝐹) ∈ On)
3 ssel2 3578 . . . . . . . . . 10 ((𝐹 No 𝑛𝐹) → 𝑛 No )
4 bdaydm 31541 . . . . . . . . . 10 dom bday = No
53, 4syl6eleqr 2709 . . . . . . . . 9 ((𝐹 No 𝑛𝐹) → 𝑛 ∈ dom bday )
6 simpr 477 . . . . . . . . 9 ((𝐹 No 𝑛𝐹) → 𝑛𝐹)
7 bdayfun 31539 . . . . . . . . . 10 Fun bday
8 funfvima 6446 . . . . . . . . . 10 ((Fun bday 𝑛 ∈ dom bday ) → (𝑛𝐹 → ( bday 𝑛) ∈ ( bday 𝐹)))
97, 8mpan 705 . . . . . . . . 9 (𝑛 ∈ dom bday → (𝑛𝐹 → ( bday 𝑛) ∈ ( bday 𝐹)))
105, 6, 9sylc 65 . . . . . . . 8 ((𝐹 No 𝑛𝐹) → ( bday 𝑛) ∈ ( bday 𝐹))
11 elssuni 4433 . . . . . . . 8 (( bday 𝑛) ∈ ( bday 𝐹) → ( bday 𝑛) ⊆ ( bday 𝐹))
1210, 11syl 17 . . . . . . 7 ((𝐹 No 𝑛𝐹) → ( bday 𝑛) ⊆ ( bday 𝐹))
13 bdayelon 31543 . . . . . . . 8 ( bday 𝑛) ∈ On
14 imassrn 5436 . . . . . . . . . 10 ( bday 𝐹) ⊆ ran bday
15 bdayrn 31540 . . . . . . . . . 10 ran bday = On
1614, 15sseqtri 3616 . . . . . . . . 9 ( bday 𝐹) ⊆ On
17 ssorduni 6932 . . . . . . . . 9 (( bday 𝐹) ⊆ On → Ord ( bday 𝐹))
1816, 17ax-mp 5 . . . . . . . 8 Ord ( bday 𝐹)
19 ordsssuc 5771 . . . . . . . 8 ((( bday 𝑛) ∈ On ∧ Ord ( bday 𝐹)) → (( bday 𝑛) ⊆ ( bday 𝐹) ↔ ( bday 𝑛) ∈ suc ( bday 𝐹)))
2013, 18, 19mp2an 707 . . . . . . 7 (( bday 𝑛) ⊆ ( bday 𝐹) ↔ ( bday 𝑛) ∈ suc ( bday 𝐹))
2112, 20sylib 208 . . . . . 6 ((𝐹 No 𝑛𝐹) → ( bday 𝑛) ∈ suc ( bday 𝐹))
22 nobndlem2.1 . . . . . . . 8 𝑋 ∈ {1𝑜, 2𝑜}
2322nosgnn0i 31513 . . . . . . 7 ∅ ≠ 𝑋
24 fvnobday 31545 . . . . . . . . 9 (𝑛 No → (𝑛‘( bday 𝑛)) = ∅)
253, 24syl 17 . . . . . . . 8 ((𝐹 No 𝑛𝐹) → (𝑛‘( bday 𝑛)) = ∅)
2625neeq1d 2849 . . . . . . 7 ((𝐹 No 𝑛𝐹) → ((𝑛‘( bday 𝑛)) ≠ 𝑋 ↔ ∅ ≠ 𝑋))
2723, 26mpbiri 248 . . . . . 6 ((𝐹 No 𝑛𝐹) → (𝑛‘( bday 𝑛)) ≠ 𝑋)
28 fveq2 6148 . . . . . . . 8 (𝑏 = ( bday 𝑛) → (𝑛𝑏) = (𝑛‘( bday 𝑛)))
2928neeq1d 2849 . . . . . . 7 (𝑏 = ( bday 𝑛) → ((𝑛𝑏) ≠ 𝑋 ↔ (𝑛‘( bday 𝑛)) ≠ 𝑋))
3029rspcev 3295 . . . . . 6 ((( bday 𝑛) ∈ suc ( bday 𝐹) ∧ (𝑛‘( bday 𝑛)) ≠ 𝑋) → ∃𝑏 ∈ suc ( bday 𝐹)(𝑛𝑏) ≠ 𝑋)
3121, 27, 30syl2anc 692 . . . . 5 ((𝐹 No 𝑛𝐹) → ∃𝑏 ∈ suc ( bday 𝐹)(𝑛𝑏) ≠ 𝑋)
3231ralrimiva 2960 . . . 4 (𝐹 No → ∀𝑛𝐹𝑏 ∈ suc ( bday 𝐹)(𝑛𝑏) ≠ 𝑋)
33 rexeq 3128 . . . . . 6 (𝑎 = suc ( bday 𝐹) → (∃𝑏𝑎 (𝑛𝑏) ≠ 𝑋 ↔ ∃𝑏 ∈ suc ( bday 𝐹)(𝑛𝑏) ≠ 𝑋))
3433ralbidv 2980 . . . . 5 (𝑎 = suc ( bday 𝐹) → (∀𝑛𝐹𝑏𝑎 (𝑛𝑏) ≠ 𝑋 ↔ ∀𝑛𝐹𝑏 ∈ suc ( bday 𝐹)(𝑛𝑏) ≠ 𝑋))
3534rspcev 3295 . . . 4 ((suc ( bday 𝐹) ∈ On ∧ ∀𝑛𝐹𝑏 ∈ suc ( bday 𝐹)(𝑛𝑏) ≠ 𝑋) → ∃𝑎 ∈ On ∀𝑛𝐹𝑏𝑎 (𝑛𝑏) ≠ 𝑋)
362, 32, 35syl2anr 495 . . 3 ((𝐹 No 𝐹𝐴) → ∃𝑎 ∈ On ∀𝑛𝐹𝑏𝑎 (𝑛𝑏) ≠ 𝑋)
37 onintrab2 6949 . . 3 (∃𝑎 ∈ On ∀𝑛𝐹𝑏𝑎 (𝑛𝑏) ≠ 𝑋 {𝑎 ∈ On ∣ ∀𝑛𝐹𝑏𝑎 (𝑛𝑏) ≠ 𝑋} ∈ On)
3836, 37sylib 208 . 2 ((𝐹 No 𝐹𝐴) → {𝑎 ∈ On ∣ ∀𝑛𝐹𝑏𝑎 (𝑛𝑏) ≠ 𝑋} ∈ On)
391, 38syl5eqel 2702 1 ((𝐹 No 𝐹𝐴) → 𝐶 ∈ On)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1480   ∈ wcel 1987   ≠ wne 2790  ∀wral 2907  ∃wrex 2908  {crab 2911   ⊆ wss 3555  ∅c0 3891  {cpr 4150  ∪ cuni 4402  ∩ cint 4440  dom cdm 5074  ran crn 5075   “ cima 5077  Ord word 5681  Oncon0 5682  suc csuc 5684  Fun wfun 5841  ‘cfv 5847  1𝑜c1o 7498  2𝑜c2o 7499   No csur 31494   bday cbday 31496 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-ord 5685  df-on 5686  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-1o 7505  df-2o 7506  df-no 31497  df-bday 31499 This theorem is referenced by:  nobndlem3  31557  nobndup  31563  nobnddown  31564
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